Using TI Interactive Software, Function Flyer or a Graphing Calculator

1. Exploring Functions Using TI Interactive Software, Function Flyer or a Graphing Calculator

2. Why Study Functions? • Before we begin our study of functions, their equations, and their graphs, we want to begin to think about how this information is actually used and what contribution it makes to our every day life.

3. Why? • In this lesson module, we will explore how determining a functional relationship that describes a real world problem (diffusion in surface water), allows us to analyze data, make predictions, and develop solutions to help prevent serious water quality issues. Tropical Storm Allison Recovery Project

4. What is a Function? A function is a relation (a set of ordered pairs) in which, for each ordered pair (x, y), the first coordinate has exactly one second coordinate. Therefore, in a function, there is a correspondence between the two variables such that each value of the first variable (x) corresponds to exactly one value of the second variable (y). Every function is a relation, but not all relations are functions.

5. Variables in a Function • The independent variable is a variable upon whose value other variables depend. • A dependent variable is a variable whose value always depends on the value of the other variables. • For example, in the equation below, p is the independent variable and q is the dependent variable.

6. Example Applied • Puppy Gourmet is sponsoring a puppy show and they need to determine the total number of participants. There will be 5 “Best of Show” winners plus 3 puppies for each Kennel represented. • Since the total number of puppies depends on the number of Kennels that participate, • p represents the number of Kennels and is the independent variable • q represents the total number of participants and is the dependent variable.

7. Functions Continued • There are three ways to describe a function: • Ordered pairs • Graph • Equation

8. Vertical Line Test • From a graph, it is easy to determine whether or not a relation is also a function by using the Vertical Line Test. • If the graph is of a function, a vertical line can be dropped from any point and it will intersect the graph at one and only one point.

9. Domain and Range • The domain of a function is the set of values that are allowable substitutes for the independent variable (the “x” values). • The range of a function is the set of values that can result from the substitutions for the independent variable. It is the set of values for the dependent variable (the “y” values). The domain of the function above is all real numbers. The range of the function above is all real numbers.

10. Identifying Parent Functions by Name, Graph, and Equation Linear Quadratic

11. Exponential Logarithmic Identifying Parent Functions Continued

12. Absolute Value Radical Identifying Parent Functions Continued

13. nth Power Function when n is an odd # nth Power Function when n is an even # Identifying Special Functions

14. Rational Polynomial Identifying More Special Functions

15. Functional Notation • Functional notation is also known as Euler’s notation. • The parentheses in this notation do not stand for multiplication. Instead, they enclose the independent variable. • An example is • It is read “f of x equals 2x – 4”. • Assignment: In your notes rewrite all the parent and special functions using functional notation.

16. Transformations of Functions • Now we are going to observe how the parent function changes as the equation changes. • To do this, we will work with the quadratic parent function first. You may use either TI Interactive Software, Function Flyer or your graphing calculator. • Your assignment is to do the Transformations Lesson. It should be completed by the end of Day 2 and turned in to your teacher.